text: First derived in 1978, Q-vector derivation can be simplified for the midlatitudes, using the midlatitude β-plane quasi-geostrophic prediction equations:
  #  (x component of quasi-geostrophic momentum equation)
  #  (y component of quasi-geostrophic momentum equation)
  #  (quasi-geostrophic thermodynamic equation)
  And the thermal wind equations:
   (x component of thermal wind equation)
   (y component of thermal wind equation)
  where  is the Coriolis parameter, approximated by the constant 1e s;  is the atmospheric ideal gas constant;  is the latitudinal change in the Coriolis parameter ;  is a static stability parameter;  is the specific heat at constant pressure;  is pressure;  is temperature; anything with a subscript  indicates geostrophic; anything with a subscript  indicates ageostrophic;  is a diabatic heating rate; and  is the Lagrangian rate change of pressure with time. .  Note that because pressure decreases with height in the atmosphere, a negative value of  is upward vertical motion, analogous to .
  From these equations we can get expressions for the Q-vector:
  And in vector form:
  Plugging these Q-vector equations into the quasi-geostrophic omega equation gives:
  If second derivatives are approximated as a negative sign, as is true for a sinusoidal function, the above in an adiabatic setting may be viewed as a statement about upward motion:
  Expanding the left-hand side of the quasi-geostrophic omega equation in a Fourier Series gives the  above, implying that a  relationship with the right-hand side of the quasi-geostrophic omega equation can be assumed.
  This expression shows that the divergence of the Q-vector () is associated with downward motion. Therefore, convergent  forces ascent and divergent  forces descend. Q-vectors and all ageostrophic flow exist to preserve thermal wind balance. Therefore, low level Q-vectors tend to point in the direction of low-level ageostrophic winds.